Question

How would I solve this trigonometric limit?

Answer 1

\[\lim_{\theta \rightarrow \Pi/2} \frac{ (1-\sin \theta) }{ (\theta-\Pi/2) }\]

Answer 2

I will use x's if you don't mind \(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~\frac{\pi}{2}}\frac{1-\sin x}{x-\frac{\pi}{2}}}\)

Answer 3

as you plug in pi/2 into top and bottom, you get 0/0
so you can take the derivative on top and bottom (i.e. apply L'Hospital's rule)

Answer 4

when you differentiate the top and bottom you get, \(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~\frac{\pi}{2}}\frac{-\cos x}{1}~~~\Rightarrow~~~~\lim_{x \rightarrow ~\frac{\pi}{2}}~-\cos(x) }\)

Answer 5

then plug pi//2,

Answer 6

Hope this is making sense

Answer 7

Does it make a difference that you replaced the thetas with 'x'?

Answer 8

well, you can imagine that theta's are x's (lol)

Answer 9

it makes no difference. you can re-write it but instead of x's use thetas

Answer 10

theta isn't so special.
it is just that I didn't feel like typing ]theta every time

Answer 11

(sorry `\theta` is the actual code )

Answer 12

Okay it makes sense thanks :)

Answer 13

:)